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arXiv · 2509.20625

The unavoidable drawings of complete multipartite graphs

Jozsef Balogh·Irene Parada·Gelasio Salazar
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Abstract

In a simple drawing of a graph every pair of edges intersect each other in at most one point, which is either a common endvertex or a proper crossing. For each positive integer $n$, Negami identified a drawing $B_n$ of the complete bipartite graph $K_{n,n}$, and proved that if $N$ is sufficiently large, then every drawing of $K_{N,N}$ contains a drawing of $K_{n,n}$ weakly isomorphic to $B_n$. Thus $B_n$ is (up to weak isomorphism) the only {\em unavoidable} drawing of $K_{n,n}$. We extend this result to complete multipartite graphs, characterizing their unavoidable drawings.

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Jozsef Balogh, Irene Parada, Gelasio Salazar. 2025-09-24. The unavoidable drawings of complete multipartite graphs. https://arxiv.org/abs/2509.20625

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